(−1)(1!2! · · · (n− 1)!) σ(nu) σ(u)n = ∣∣∣∣∣∣∣ ℘ ℘ · · · ℘ ℘ ℘ · · · ℘ .. .. . . . .. ℘ ℘ · · · ℘ ∣∣∣∣∣∣∣ (u). (0.2) Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 1 2℘ (u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the fu...

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

We derive a general expression for an interface parameter which makes possible the design of a neutral elliptic inhomogeneity when the stress field in the surrounding matrix is a polynomial function of nth order and the composite is subjected to antiplane shear deformations. © 2005 Elsevier Ltd. All rights reserved.

Denote C (d) ∆ the relative symmetric product of this family over ∆ , parameterizing effective divisors of degree d on fibres Ct for t 6= 0. We will construct a compactification H̃d of C (d) ∆ over ∆, such that H̃d has smooth total space and the fibre over t = 0 has simple normal crossing support. By studying the fibre over t = 0 of H̃d, we understand how the symmetric products of smooth curves de...

To help motivate the Weil pairing, we discuss it in the context of elliptic curves over the field of complex numbers.

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

The study of elliptic curves has historically been a subject of almost purely mathematical interest. However, Koblitz and Miller independently showed that elliptic curves can be used to implement cryptographic primitives [13], [17]. This thrust elliptic curves from the abstract realm of pure mathematics to the preeminently applied world of communications security. Public key cryptography in gen...

Three improvements in reduction and computation of elliptic integrals are made. 1. Reduction formulas, used to express many elliptic integrals in terms of a few standard integrals, are simplified by modifying the definition of intermediate "basic integrals." 2. A faster than quadratically convergent series is given for numerical computation of the complete symmetric elliptic integral of the thi...

We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic ...

Scalar multiplication on Legendre form elliptic curves can be speeded up in two ways. One can perform the bulk of the computation either on the associated Kummer line or on an appropriate twisted Edwards form elliptic curve. This paper provides details of moving to and from between Legendre form elliptic curves and associated Kummer line and moving to and from between Legendre form elliptic cur...